(a) Technical Field
The present disclosure relates in general to numerical analysis or applied mathematics. More particularly, it relates to a solution-finding method which finds an approximate solution of an equation in commercial software required for calculation in mathematics and engineering, and a numerical analysis code using the solution-finding method.
(b) Background Art
Generally, when the shape of a gas turbine is designed or when the appearance of an airplane or a vehicle is designed, an optimal shape is designed using hydrodynamics. With the development of computers, numerical analysis using hydrodynamics has been mainly conducted using computers, and solutions (or roots) of a large number of equations related to hydrodynamics are generally obtained by finding approximate solutions.
Conventional methods of finding approximate solutions may include, for example, a bisection method, Newton's method, and a secant method. A bisection method is a method using the change of the sign of a function in an interval so as to find a solution present in the interval, and is configured to obtain a solution by always bisecting an interval and finding a section in which the sign of a function changes. Such a bisection method is advantageous because a solution can always be obtained, but is disadvantageous because the speed of convergence is low.
Newton's method is a method of obtaining a solution using a derivative, and is advantageous because the speed of convergence is very high near a solution. However, there is a problem in that when an equation has a slope close to ‘0’, it is difficult to obtain a solution, and in that when an initially estimated value is erroneously obtained, Newton's method does not converge on a solution, thus making it impossible to find a solution.
A secant method is a method obtained by modifying Newton's method into a two-point method so as to solve the problem of calculating a derivative in Newton's method. However, this method is problematic in that the speed of convergence is generally slightly lower than that of Newton's method, two initial values are required, and the probability of failing in convergence according to the behavior of a function cannot be excluded.
As described above, the conventional solution-finding methods are problematic in that the speed of convergence is excessively low and the speed of calculation is low, and in that, when an initially estimated value is erroneously set, it is impossible to find a solution, or, alternatively, the time of calculation required to find a solution increases.
The above information disclosed in this Background section is only for enhancement of understanding of the background of the invention and therefore it may contain information that does not form the prior art that is already known in this country to a person of ordinary skill in the art.